Classial Diamagnetism and the Satellite Paradox 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (November 1, 008) In typial models of lassial diamagnetism (see, for example, se. 34-4 of [1] or se. 11.5 of []) the hange of the magneti moment of an atom is dedued when an external magneti field inreases slowly from 0 to B = B ẑ. The atom onsists of an eletron of harge e and mass m that is in a irular orbit of radius r 0 in the x-y plane about a fixed nuleus of harge e at the origin. It is taitly assumed that the radius of the orbit remains r 0 at all times, although this is not onsistent with the effet of perturbations on motion in a 1/r potential, even if radiation is ignored (as it must be in any lassial model of a stable atom). 1 Give a model of lassial diamagnetism that inludes the (small) effet of hanges in the radius of the orbit as external magneti field inreases. Compare the present ase to the so-alled satellite paradox [4, 5, 6]. The latter is that the effet of atmospheri drag on a satellite in a low orbit about the Earth is to inrease the speed of the satellite as it slowly spirals inwards towards the Earth s surfae. Solution Aspets of this problem have been disussed in [7]. We suppose that at time t = 0 an external magneti field, perpendiular to the (x-y) plane of the eletron s orbit, turns on aording to B(t) =Ḃt (t >0), (1) where Ḃ is a onstant. As a result, an external azimuthal eletri field, E ext (r, t) = rḃ ˆφ (t >0), () is generated aording to Faraday s law (in ylindrial oordinates (r, φ, z) withthe fixed nuleus at the origin, and in Gaussian units). Then, the external (Lorentz) fore on the eletron is ( F ext = e E ext + v ) B ext = erḃ ˆφ ev φb ˆr + eṙb ˆφ, (3) 1 Diamagnetism has reently been onsider in another type of lassial atom [3], onsisting of an eletron somehow onfined to the surfae of a sphere. Faraday s law atually only tells us that E φ rdφ = πr Ḃ/, sothat E φ = rḃ/ where the average is taken over the orbit. In general, the indued eletri field inludes another omponent in the x-y plane that is essentially uniform over the atom, whih has the effet of reating a small eletri dipole moment for the atom while the magneti field is hanging. We ignore this tiny effet, whih in any ase vanishes one the magneti field takes on a steady, nonzero value. 1
where v =ṙ ˆr + r φ ˆφ, andwewriter φ = v φ.asv φ B = v φ tḃ, the first and seond terms of eq. (3) are of similar order (and both are small ompared to the Coulomb fore ee ˆr/r ). We suppose that the hange in magneti field is so slow that the resulting radial veloity ṙ is negligible ompared to the azimuthal veloity v φ ; hene, the third term is small ompared to the first and seond terms of eq. (3). However, the third term plays a role in insuring that a magneti field does no work. Thus, during time dt the eletron moves distane and the work done on it by the external fields is ds = v dt =(ṙ ˆr + r φ ˆφ) dt (4) dw ext = F ds = e v E ext dt = er φḃ dt = erv φḃ The equation of motion for the eletron is ma = m ( r r φ ) ˆr + m ( r φ +ṙ φ ) ( ˆφ = e E tot + v ) B ext dt. (5) = ee erḃ ˆr + r ˆφ ˆr + eṙb ˆφ, (6) For slow hanges in the magneti field we neglet r ompared to r φ, so the radial equation of motion an be written as mr φ ee r + = mr3 0 φ 0 r +, (7) where r 0 and v 0 = r 0 φ0 are the radius and azimuthal veloity, respetively, of the irular orbit when B =0. The azimuthal equation of motion is m(r φ +ṙ φ) = erḃ + eṙb. (8) The terms r φ and ṙ φ are of omparable magnitude. Multiplying eq. (8) by r, we an integrate to find the (mehanial) angular momentum, Using eq. (9) for φ in eq. (7), we find L z = mr φ = mr 0 φ0 + er B. (9) r0 3 r r 0 3 r e rb. (10) 4m r 0 φ 0 Assuming a solution of the form r = r 0 (1 + Δ), we obtain r r 0 1 e B ( 4m φ = r 0 1 e r0b ). (11) 4m v 0 0
That the radius r of the perturbed orbit departs from r 0 by a term that is seond order in the magneti field B was mentioned in [7] without a detailed derivation. Returning to eq. (9), the angular veloity is, to seond order in B, φ φ 0 + eb m + e B m φ, (1) 0 and the azimuthal veloity is v φ v 0 + er 0B m + e r0 B, (13) m v 0 to the same order. The motion (11)-(13) does not have the harater of that in the satellite paradox. For positive v 0, the energy of the lassial atom is inreased by the perturbing magneti field. That is, W ext > 0 aording to eq. (5), while the radius of the orbit derease with time aording to eq. (11), and the azimuthal veloity inreases aording to eq. (13). This is the behavior for negative, rather than positive, Ẇ ext in the satellite paradox [4, 5]. To first order in the magneti field B, eq. (13) is the same as obtained by supposing that the azimuthal fore ee φ diretly affets the azimuthal veloity (i.e., thateq.(8)anbe approximated as v φ = er 0 Ḃ/m), in ontrast to the ase of the satellite paradox where the hange in azimuthal veloity is the negative of the naive expetation [4]. However, eq. (13) differs in seond order from the result of this assumption, as needed to explain the energy balane to this order (see Appendix A). Indeed, if one is willing to aept eq. (13) to first order as obvious, then the radial equation of motion an be written as mv φ r ee r + ev φb = mr 0v 0 r + ev φb whih is a quadrati equation in 1/r with the approximate solution, (14) 1 r 1 e r 0 B, (15) r 0 4m v0 whih leads to eq. (11) but with the opposite sign for the seond term. Coming at length to the issue of diamagnetism, we note that the orbital angular momentum is L = mrv φ ẑ, (16) The magneti moment of the system is, following Larmor, μ = e m L = erv φ ẑ erv 0 ẑ e r0 4m B+ e3 r0b 3 ẑ μ 8m 3 v 0 +μ diamagneti +O(B ), (17) 0 where μ 0 = erv 0 ẑ, and μ diamagneti = e r0 4m B (18) as in the usual treatments [1, ] whih simply assume that r = r 0 always. From eq. (11) we see that this is a very good assumption within the ontext of a lassial atom (so long as one ignores effets of radiation). 3
A Appendix: Energy Balane The energy of the lassial atom in a magneti field B = B(t) ẑ is 3 U = m(ṙ + v φ ) We ignore the ontribution of ṙ to the kineti energy, and write U mv φ mr 0v0 ( mv 0 1+ er 0B + e r0b ) mv r mv 0 4m v0 0 = mv 0 + er 0v 0 B to seond order in field strength B. Then, to this order, U er 0v 0 Ḃ realling eqs. (5) and (13). + e r 0BḂ 4m ee r. (19) ( 1+ e r 0B 4m v 0 + e r 0B 8m (0) = er 0v 0 Ḃ ( 1+ er ) 0B erv φḃ mv 0 ) = Ẇext, (1) B Appendix: Why Doesn t the Satellite Paradox Hold for Classial Diamagnetism? One answer is that magneti fields do no work. The phenomenon of lassial diamagnetism, as desribed by eqs. (11) and (13), ontradits the laim in [5] that the qualitative behavior seen in the satellite paradox (the veloity inreases as the radius dereases, and vie versa) holds for any perturbation of motion in a 1/r potential. The argument in [5] is based on the laim that all relevant behavior of the system an be related to its energy, and that hanges in the energy are due to the work done by the perturbing fores. However, sine magneti fields do no work, they an lead to perturbations whose effet is not aptured by arguments based on energy. For ompleteness, we analyze the satellite paradox in a manner similar to that given in se.. Suppose the external fore is purely a drag fore, F ext = Cv p v Cv p φ (ṙ ˆr + v φ ˆφ), () where p is a small non-negative number, perhaps 0 or 1, and C is the drag oeffiient. Unlike the ase of an atom in an external magneti field, it suffies to onsider only the ase that v φ is positive. We again neglet r ompared to r φ in the radial equation of motion, φ φ 0 r3 0 r 3 Cṙvp φ mr, or v φ v 0 r 0 Crṙvp φ r m, (3) 3 Note that the energy, μ B = ervb/ er 0 v 0 B/, of the magneti moment μ in the external magneti field B is already ontained in the mirosopi energy expression (19), whih is also the energy in the Darwin approximation (see, for example, eq. () of [8]). 4
where v 0 is the azimuthal veloity of the (nearly) irular orbit at some referene radius r 0. The effet of the drag fore is to derease r, sothatṙ is negative, and eq. (3) tells us that v φ >v 0 for all r<r 0. Thus, the radial equation of motion ontains the qualitative nature of the satellite paradox, that the drag fore inreases the azimuthal veloity, rather than reduing it. In the first approximation, we ignore the small radial omponent of the drag fore, so that φ φ 0r0 3 ( ) r0 1/, and v r 3 φ v 0. (4) r The azimuthal equation of motion is r φ +ṙ φ Cr1+p φ1+p =. (5) m An analyti solution is espeially simple for p = 0 (drag fore proportional to veloity). 4 As before, we use the derivative of the result (4) of the radial equation in the azimuthal equation (5) to find ṙ = C r, (6) m and hene r = r 0 e Ct/m. (7) Thus, using eq. (4), v φ = v 0 e Ct/m. (8) In ontrast, the radius (11) of the lassial atom in an external magneti field dereases as the field inreases whether the azimuthal fore applies a drag or a boost! Furthermore, when the azimuthal fore on the lassial atom is a drag the azimuthal veloity dereases in magnitude, and inrease when the fore is a boost. C Appendix: Adiabati Invariane A more formal approah to the problem of lassial diamagnetism an be based on the (nonrelativisti) Lagrangian, 5 L = mv ev A + ev = m(ṙ + r φ ) (t) + ee r, (9) for the eletron (of harge e) in the Coulomb potential V = e /r of the nuleus and the external magneti field B(t) =B ẑ = A where A = rb ˆφ/. As the Lagrangian (9) does not depend on oordinate φ, the anonial momentum p φ is onserved, and we identify this as the z omponent of the anonial angular momentum, l z = p φ = L φ = er mr φ B = L z er B = L 0 = mr 0 v 0, (30) 4 An analyti solution is also possible for the ase of p = 1, i.e., for a onstant drag fore [6]. 5 In the present example the term ev A/ an be rewritten as μ B, whereμ is the (orbital) magneti moment. 5
where L z = mr φ is the mehanial angular momentum about the z axis. Thus, φ = L 0 mr + eb m, (31) where eb(t)/m is the ylotron (Larmor) frequeny. The radial equation of motion is m r = mr φ ee r = L 0 mr 3 e rb 4m L 0 mr 0 r U eff r, (3) wherewenotethatee = L 0/mr 0 for the initial irular orbit. The effetive radial potential U eff is U eff (r, t)= L 0 mr + e r B L 0 8m mr 0 r. (33) However, the problem is not one of small osillations about a fixed minimum of the effetive potential. Rather, we suppose that the orbit at time t is nearly irular with radius r(t) suh that the effetive potential is minimum at this time. That is, we set eq. (3) to zero, whih leads again to eq. (10), to the solutions (11)-(13) and to the magneti moment (18). We have found one invariant in this problem, the anonial angular momentum l z of eq. (30). Beause this remains onstant during hanges in the magneti field, whether these are adiabati or not, we an ertainly identify the anonial angular momentum as an adiabati invariant. We ould define the anonial magneti moment μ C as μ C = e m l z = μ z + e r B m μ z + e r0b m, (34) whih is also an adiabati invariant. However, the ordinary magneti moment μ z is not an adiabati invariant in this problem. We an ompare the present ase of a lassial atom to that of an eletron in a uniform magneti field. For bound motion in the latter ase we annot begin with zero magneti field, but we suppose instead that B = B 0 ẑ at time t =0. Then, the Lagrangian is obtained from eq. (9) by setting e =0. The anonial angular momentum (30) is again (an adiabati) invariant, while the radial equation of motion simplifies to mr φ = er B/ for slow hanges in the magneti field B. Hene, the anonial angular momentum l z an be written as l z = mr er φ B and the anonial magneti moment is = er B = L z, (35) μ C = e m l z = μ z. (36) In this ase the ordinary magneti moment μ z is also an adiabati invariant. However, most disussions of the motion of harged partiles in slowly varying magneti fields do not emphasize that the ordinary magneti moment is (somewhat aidentally) twie the invariant anonial magneti moment, whih may leave a misimpression as to the (non)invariane of the ordinary magneti moment in other irumstanes. 6
Aknowledgment Thanks to Vladimir Onoohin for drawing the author s attention to this problem, and to Vladimr Hnizdo and Dmitry Peregoudov for e-disussions of it. Referenes [1] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Letures on Physis, Vol. (Addison-Wesley, Reading, MA, 1964). [] E.M. Purell, Eletriity and Magnetism, nd ed. (MGraw-Hill, New York, 1985). [3] On Non-zero Classial Diamagnetism: A Surprise, N. Kumar and K.V. Kumar (Nov. 0, 008), http://puhep1.prineton.edu/~mdonald/examples/em/kumar_physis-lass-ph-0811-3071.pdf [4] B.D. Mills, Jr., Satellite Paradox, Am. J. Phys. 7, 115 (1959), http://puhep1.prineton.edu/~mdonald/examples/mehanis/mills_ajp_7_115_59.pdf [5] L. Blitzer, Satellite Orbit Paradox: A General View, Am. J. Phys. 39, 887 (1971), http://puhep1.prineton.edu/~mdonald/examples/mehanis/blitzer_ajp_39_887_71.pdf [6] F.P.J. Rimrott and F.A. Salustri, Open Orbits in Satellite Dynamis, Teh.Meh.1, 07 (001), http://puhep1.prineton.edu/~mdonald/examples/mehanis/rimrott_tm_1_07_01.pdf [7] S.L. O Dell and R.K.P. Zia, Classial and semilassial diamagnetism: A ritique of treatment in elementary texts, Am. J. Phys. 54, 3 (1986), http://puhep1.prineton.edu/~mdonald/examples/em/odell_ajp_54_3_86.pdf [8] K.T. MDonald, Darwin Energy Paradoxes (Ot. 9, 008), http://puhep1.prineton.edu/~mdonald/examples/darwin.pdf 7